Symbolic dynamics for geodesic flow on CAT(-1) spaces

Penn State Dynamical Systems Workshop, Pesin 70th Birthday Event, October 2017

Mathematical Congress of the Americas, Montreal (Canada), July 2017

The geodesic flow on a compact locally CAT(-1) metric space, first studied by Gromov, is a far-reaching generalization of the geodesic flow on a closed negative curvature Riemannian manifold. While one expects these flows to exhibit similar behaviour to the classical case, the lack of smooth structure has been a major obstacle to extending many of the finer aspects of the dynamical theory to this setting. Our new approach to this problem is to show that such geodesic flows are Smale flows.

A Smale flow is a topological flow equipped with a continuous bracket operation which is an abstraction of the local product structure from uniform hyperbolicity. In 1987, Pollicott showed that a version of Bowens construction of symbolic dynamics for Axiom A flows can be extended to this setting. By symbolic dynamics, we mean there exists a suspension flow over a shift of finite type which describes the original dynamics. By taking additional care in the construction, we are able to verify that the roof function can be taken to be Lipschitz in our setting. This is achieved by using carefully chosen geometric rectangles as the building blocks for the construction.

With this additional ingredient, the symbolic dynamics machine switches on and ergodic- theoretic results which are true for Axiom A flows are extended to this setting. For example, we obtain that the Bowen-Margulis measure for the geodesic flow is Bernoulli and satisfies the Central Limit Theorem. This is joint work with Dave Constantine and Jean-Francois Lafont.

Uniqueness of equilibrium states for geodesic flows in manifolds of nonpositive curvature

Geometry seminar, Yale, March 2016

Dynamics seminar, Michigan, February 2016

We establish results on uniqueness of equilibrium states for geodesic flows on rank one manifolds. This is an application of machinery developed by Vaughn Climenhaga and myself, which applies when systems satisfy suitably weakened versions of expansivity and the specification property. The geodesic flow on a rank one manifold is a canonical example of a non-uniformly hyperbolic flow and I'll explain why it satisfies our hypotheses. Our methods are completely different from those used by Knieper in his seminal proof that there is a unique measure of maximal entropy in this setting. This is joint work with Keith Burns (Northwestern), Vaughn Climenhaga (Houston) and Todd Fisher (Brigham Young).

Entropy for generalised beta-transformations

British Mathematics Colloquium, Queen Marys (London), April 2014

Pure Mathematics Colloquium, St. Andrews, April 2014

Generalised beta-transformations are the class of piecewise continuous interval maps given by taking the beta-transformation x↦βx (mod1), where β > 1, and replacing some of the branches with branches of constant negative slope. We would like to describe the set of beta for which these maps can admit a Markov partition. We know that beta (which is the exponential of the entropy of the map) must be an algebraic number. Our main result is that the Galois conjugates of such beta have modulus less than 2, and the modulus is bounded away from 2 apart from the exceptional case of conjugates lying on the real line. This extends an analysis of Solomyak for the case of beta-transformations, who obtained a sharp bound of the golden mean in that setting.

I will also describe a connection with some of the results of Thurston's fascinating final paper, where the Galois conjugates of entropies of post-critically finite unimodal maps are shown to describe an intriguing fractal set. These numbers are included in the setting that we analyze.

Equilibrum states for certain robustly transitive diffeomorphisms

Ergodic theory and dynamical systems seminar, University of Warwick, April 2014

We establish results on uniqueness of equilibrium states for the well-known Mane examples of robustly transitive diffeomorphisms. This is an application of machinery developed by Vaughn Climenhaga and myself, which applies when systems satisfy suitably weakened versions of expansivity and the specification property. The Mane examples are partially hyperbolic maps of the 3-torus, and I'll explain why these maps satisfy our hypotheses. This is joint work with Vaughn Climenhaga (Houston) and Todd Fisher (Brigham Young).

Coding Sequence Density Estimation via Topological Pressure

ICIAM conference, Ohio State MBI, May 2014

I will describe an approach to coding sequence (CDS) density estimation in genomic analysis introduced recently by myself and David Koslicki. Our approach is based on the topological pressure, which is a measure of ‘weighted information content’ adapted from ergodic theory. We use the topological pressure (with suitable training data) to give ab initio predictions of CDS density on the genomes of Mus Musculus, Rhesus Macaque and Drosophila Melanogaster. While our method is not sufficiently precise to predict, for example, the exact locations of genes, we demonstrate that our method gives reasonable estimates for the ‘coarse scale’ problem of predicting CDS density. This is joint work with David Koslicki (Oregon State).

Large deviations and horseshoes for S-gap shifts.

IUPUI, October 2013

This is joint work with Vaughn Climenhaga and Kenichiro Yamamoto. I will explain how we establish the large deviations principle for the family of S-gap shifts. Key to our approach is a 'horseshoe theorem' which allows us to approximate invariant measures by ergodic measures supported on sofic subshifts of the S-gap shift. Although I will focus on the concrete example of S-gap shifts for ease of exposition, our large deviations result actually applies much more generally, and is based on machinery developed by Climenhaga and myself to prove uniqueness of equilibrium states in various different 'non-uniform' settings.

Intrinsic ergodicity, orbit gluing and S-gap shifts

Midwest Dynamics Conference, October 2013

This talk is based on a series of papers by Vaughn Climenhaga (Houston) and myself where we develop a new approach to prove uniqueness of equilibrium measures for dynamical systems with various non-uniform structures.

One class of model examples that motivated our results is the family of S-gap shifts. The S-gap shifts are a natural family of symbolic spaces which are easy to define but can be challenging to study! For a fixed subset S of the natural numbers, the corresponding S-gap shift is the collection of binary sequences which satisfy the condition that the length of every run of consecutive 0's is a member of S. Examples are the even shift (we let S be the even numbers) and the prime gap shift (we let S be the prime numbers).

The key difficulty in the study of S-gap shifts is a spectacular failure of the Markov property, and we develop techniques to deal with this. These techniques can be adapted to apply to many other interesting dynamical systems beyond the well understood uniformly hyperbolic case (e.g. beta-shifts, interval maps with parabolic fixed points, non-uniformly expanding maps in higher dimensions, some partially hyperbolic examples).

I will explain our approach and motivations focusing on the example of S-gap shifts, building up to our result that 'Every subshift factor of an S-gap shift (or a beta-shift) is intrinsically ergodic'. If time permits, I will also describe a brand new result by Climenhaga, myself and Kenichiro Yamamoto (Tokyo Denki University) which establishes a large deviations principle for S-gap shifts.

A 'horseshoe' theorem in symbolic dynamics via single sequence techniques.

Mathematical Congress of the Americas, August 2013

For a broad class of symbolic dynamical systems without the Markov property, including the coding spaces of many piecewise continuous interval maps, we show how to approximate an arbitrary ergodic measure with a measure of almost the same entropy supported on a sofic subshift. This is interpreted as a symbolic analogue of a `hyperbolic horseshoe' theorem. In addition to the intrinsic interest of this result as a structure theorem, it can be a useful tool in large deviations theory and multifractal analysis. I will discuss two ways to establish this result, both based on surgery on a single generic orbit. One proof is based on Ornstein's d bar metric, and the other is based on the theory of Kolmogorov complexity. Both techniques can be explained in a simple and intuitive way.

Equilibrium states and large deviations principles for beta-shifts, S-gap shifts, and beyond.

Conference on Thermodynamic Formalism, PUC (Chile), July 2013

I will give an overview, and report on recent progress, for a long-term project joint with Vaughn Climenhaga concerning measures of maximal entropy and equilibrium states for a large class of dynamical systems with a 'non-uniform orbit structure', including piecewise continuous and parabolic interval maps, and some higher dimensional partially hyperbolic examples.

I will explain our approach and motivations focusing on the illustrative example of S-gap shifts, building up to our result that 'the level 2 large deviations principle holds for every S-gap shift', which was recently obtained by Climenhaga, myself and Kenichiro Yamamoto. If time permits, I will also describe our application to partially hyperbolic examples which is joint work with Todd Fisher.

Uniqueness of equilibrium states: Beta-shifts, the Bowen property and Beyond

Maryland-Penn State Workshop on Dynamical Systems, Maryland, April 2011

Dynamics seminar, CUNY, May 2011

This joint work with Vaughn Climenhaga (Maryland) establishes uniqueness of equilibrium states for

1) a large class of shift spaces which includes every beta-shift;

2) a large class of potential functions which strictly includes those with the Bowen property.

As an application, our method yields new results in the theory of thermodynamic formalism for piecewise monotonic interval maps. Our method allows us to handle a variety of systems without a Markov structure, and it covers a class of potentials that are well behaved away from a 'small' set; for example, an indifferent fixed point or a point of discontinuity. This work extends the techniques which we developed in a recent preprint, available at http://arxiv.org/abs/1011.2780, which gave a positive answer to the question 'Is every subshift factor of a beta-shift intrinsically ergodic?'. This question was included in Mike Boyle's article 'Open problems in symbolic dynamics', and was the original motivation for the development of these techniques.

Distinguished invariant measures for Beta-shifts and their factors.

University of Washington, Seattle, April 2011

This talk is based on joint work with Vaughn Climenhaga (Maryland), in which we show that every shift space which is a factor of a beta-shift has a unique measure of maximal entropy. This provides an affirmative answer to Problem 28.2 of Mike Boyle's article 'Open problems in symbolic dynamics'.

A measure of maximal entropy is a measure which witnesses the greatest possible complexity in the orbit structure of a topological dynamical system. Establishing when a system has a unique measure of maximal entropy is a fundamental topic in ergodic theory and has been studied extensively since the 1960's. The beta-shifts are a class of symbolic spaces with an extremely rich structure and a profound connection to number theory.

Our method actually applies to a rather large general class of shift spaces, and can also be applied to non-symbolic systems. We have recently extended our results to establish uniqueness of equilibrium measures for a large class of potential functions. We obtain new results in the setting described above, and even in the case of the full shift. I will give a detailed explanation of the problems described above and their motivation. I will also describe, via a detailed description of the beta-shift, the key ideas behind our method.

'Subshift factors of the beta-shift are intrinsically ergodic'

University of Richmond (2010 AMS Fall Southeastern meeting), November 2010

Penn State Workshop in Dynamical Systems and Related Topics, October 2010

This talk is based on joint work with Vaughn Climenhaga (Maryland), in which we show that every subshift factor of a beta-shift has a unique measure of maximal entropy. This provides an affirmative answer to Problem 28.2 of Mike Boyle's article 'Open problems in symbolic dynamics'. I'll explain the problem and its relation to existing results and give an idea of how our approach works. Our techniques are essentially new, and allow us to deal with dynamical systems which are a long way from being Markov. I will describe a broad and natural class of shift spaces to which our results apply.

'Topological pressure for non-compact sets, Kamae entropy and Kolmogorov complexity'

Logic Seminar, Pennsylvania State University, November 2010

For compact spaces, the theory of topological pressure and equilibrium states is a cornerstone of modern ergodic theory. It has long been thought desirable to generalize this theory to non-compact spaces. I will give a brief over-view of the various approaches to this problem in the literature and give an elementary alternative definition of topological pressure in the non-compact setting. It turns out that this definition is a generalization of the Kamae entropy. The definition assigns a non-negative number to each point in the space and can be interpreted as a sort of complexity. In Cantor space, this quantity is related to the Kolmogorov complexity. I will attempt to give an accessible explanation of the dynamical systems context of this theory. I will then describe some results by Brudno, Kamae, Van Lambalgen and White to illustrate the connection with Kolmogorov complexity and to demonstrate why this may be of interest to Logicians.

'Subshift factors of the beta-shift are intrinsically ergodic'

Penn State Dynamics conference October 2010, South-Eastern AMS meeting November 2010

This talk is based on joint work with Vaughn Climenhaga, in which we show that every subshift factor of a $\beta$-shift has a unique measure of maximal entropy. This provides an affirmative answer to Problem 28.2 of Mike Boyle's article `Open problems in symbolic dynamics'. I'll explain the problem and its relation to existing results and give some idea of how our approach works. I'll also discuss some other examples of symbolic spaces where our technique can be applied.

'A criterion for topological entropy to decrease under normalised Ricci flow'

Versions of this talk were given at Penn State, Yale, Northwestern, Maryland and Rice

In 2004, Manning showed that the topological entropy of the geodesic ﬂow for a surface of negative curvature decreases as the metric evolves

under the normalised Ricci ﬂow. It is an interesting open problem,also due to Manning, to determine to what extent such behaviour persists for higher

dimensional manifolds. I will describe a curvature condition on the metric under which monotonicity of the topological entropy can be

established for a short time. In particular, this criterion applies to metrics of negative sectional curvature which are in the same

conformal class as a metric of constant negative sectional curvature.

'Interpretation and applications of topological entropy in geometry'

Penn State, Working Geometry seminar, February 2010

Interpretation: The topological entropy is one of the key invariants in the theory of dynamical systems. When the dynamical system is a geodesic flow on a negatively curved manifold, it is good to know that the topological entropy has a very natural formulation as a geometric quantity. It is the exponential growth rate of volume in the universal cover. I'll sketch the classic proof of this fact which is due to Manning. Application: I'll sketch A. Katok's classic proof of his fascinating result which tells us that the topological entropy can be used to characterize which metrics on a surface are hyperbolic. More precisely, the topological entropy of the geodesic flow is minimised at the constant curvature metrics.

'The irregular set for the beta transformation carries full topological entropy (= log beta) and full Hausdorff dimension (=1)'

Penn State, Dynamical Systems seminar, October 2009, Boston University, Dynamical Systems seminar, October 2009

The beta-transformation f(x) = beta x (mod 1) has been widely studied since its introduction by Renyi in 1957. The sustained interest in the study of the beta-transformation arises from its connection with number theory and its special role as a model example of a one-dimensional expanding dynamical system which admits discontinuities.

We show that for the beta transformation, the set of points for which the Birkhoff average of a continuous function does not exist (which we call the irregular set) is either empty or has full topological entropy and Hausdorff dimension.

This result follows from a corresponding result about dynamical systems which satisfy a topological dynamical property which we call almost specification. It turns out that every beta-shift satisfies almost specification, and we will explain how this works.

'Another look at topological pressure for non-compact sets'

Seminar (part of themed semester in ergodic theory), University of Surrey, March 2009

It has long been thought desirable to generalise the standard theory of topological pressure and equilibrium states to non-compact spaces. Notably, Sarig developed the theory of Gurevic pressure for countable state shifts. This theory has many applications, particularly in the study of non-uniformly hyperbolic systems, where inducing schemes lead naturally to the study of countable state shifts.

In another direction, Bowen defined topological entropy for non-compact sets as a characteristic of dimension type. The study of topological entropy for the multifractal decomposition of Birkhoff averages (for example) is a well accepted goal in its own right. Pesin and Pitskel contributed a definition of topological pressure for non-compact sets which generalises the Bowen definition.

We give an elementary alternative definition of topological pressure in the non-compact setting. The definition is made via a suitable variational principle leading to an alternative definition of equilibrium state. We derive some properties of the new topological pressure and compare it with the other definitions. We give a simple example which illustrates the difference in the thermodynamic properties of the new pressure and the Pesin and Pitskel pressure. We describe an example taken from the multifractal analysis of the Lyapunov exponent for the Manneville-Pomeau family of maps, which seems particularly well adapted to our new framework. We hope that the new theory will have useful applications and we mention some aspirations in this direction.

'The irregular set for the beta transformation carries full topological entropy (= log beta)'

London Mathematical Society one day ergodic theory meeting, University of Warwick, January 2009;

Dynamical Systems Seminar, Northwestern University, October 2008

A recent weakening of the specification property provides new tools to study interesting systems beyond the scope of uniformly hyperbolic dynamics such as the beta-transformation. This property was introduced by Pfister and Sullivan as the g-almost product property. The version we study is a priori slightly weaker and we rename it the almost specification property. We show that for dynamical systems with almost specification, the set of points for which the Birkhoff average of a continuous function does not exist (which we call the irregular set) is either empty or has full topological entropy.

Every beta-shift satisfies almost specification and we show that the irregular set for any beta-shift or beta-transformation is either empty or has full topological entropy and Hausdorff dimension.

The talk is in three parts. Firstly, we give some history on results about the topological entropy and Hausdorff dimension of the irregular set. Secondly, we discuss our abstract results and try to give some intuition as to why they are true. Thirdly, we discuss in detail the application to the beta-transformation. In particular, we give some intuition on the almost specification property and show why it is satisfied by the beta-transformation.

'The Liouville entropy of a 3-manifold is not monotonic along the Ricci flow'

Maryland-Penn State Workshop on Dynamical Systems, Penn State, October 2008

In 2004, Manning used an important formula of Katok, Knieper and Weiss to prove that as the metric on a negatively curved surface evolves under the (normalised) Ricci flow, the topological entropy of the geodesic flow decreases. In contrast, we observe that an example of Flaminio can be used to show that the Liouville entropy can either increase or decrease along a Ricci flow in a neighbourhood of a particular 3-manifold of constant negative curvature. We will introduce this topic, explain our observation, and recall some of the interesting and challenging open questions in this area.

'The Irregular Set for Maps with the Specification Property Carries Full Topological Pressure'

Maryland-Penn State Workshop on Dynamical Systems, University of Maryland, March 2008

We describe a result that applies to any dynamical system (X, T) with the specification property. Systems satisfying specification include any continuous map which is a factor of a topologically mixing shift of finite type and any topologically mixing continuous interval map. We show that, for a generic function f on X, the irregular set of f (the set of points for which the Birkhoff average of f does not exist) carries full topological pressure (in the sense of Pesin and Pitskel). Topological pressure is interpreted as a ‘weighted’ dynamical size, so the result says that the irregular set is as ‘large’ as it can be in an appropriate topological sense. This result may be surprising given that the irregular set has zero measure with respect to any invariant measure. In the case of topological entropy, this phenomenon was first noticed for Bernoulli shifts by Pesin and Pitskel and generalised to a variety of uniformly hyperbolic systems by Barreira and Schmeling. As an application, we show that the irregular set for a suspension flow over a continuous map with specification carries full topological entropy, generalising a result of Barreira and Saussol.

'A Thermodynamic Definition of Topological Pressure for Non-Compact Sets'

Dynamics Seminar, Warwick Mathematics Institute, January 2008; Dynamics Seminar, Manchester University, November 2007

In the 80's, Pesin and Pitskel defined topological pressure for non-compact sets as a characteristic of dimension type, generalising a definition of topological entropy introduced by Bowen. We give an alternative definition of topological pressure in the non-compact setting via a suitable variational principle. We derive some properties of the new topological pressure and compare them with the properties of the Pesin and Pitskel pressure. We describe a simple example which illustrates the difference in the thermodynamic properties of the two quantities. We conclude with an example taken from the multifractal analysis of the Lyapunov exponent for the Manneville-Pomeau family of maps, which seems particularly well adapted to our new framework.

'The Irregular Set for Maps with the Specification Property carries Full Entropy'

Conference ‘Chaotic Properties of Dynamical Systems’, Warwick Mathematics Institute, August 2007

The content was similar to 'The Irregular Set for Maps with the Specification Property Carries Full Topological Pressure' (but less general).

'The Multifractal Miracle: Multifractal Analysis in Symbolic Dynamics'

Postgraduate Seminar, Warwick Mathematics Institute, November 2006

This was a talk for a general postgraduate audience.

I offer two abstracts for the talk, the second being of a more technical and precise nature:

Abstract Version 1: We are interested in the average value of a function along the orbit (under a continuous transformation) of a point. This quantity is known as the Birkhoff (or ergodic) average. Does this average exist? What values does it take? It has been known since the 1930's that from the point of view of measure theory the Birkhoff average is very well behaved. However, from the topological viewpoint, a priori, this average behaves rather badly. However, by the so called 'multifractal miracle', one can say a surprising amount about the destination of the Birkhoff sum. In the context of symbolic dynamics, I will describe the 'multifractal miracle' in a way accessible to a general postgraduate audience.

Abstract Version 2: Birkhoff's ergodic theorem tells us that for an integrable function f on a dynamical system, the Birkhoff sum converges to $\int f d m$ almost everywhere with respect to an ergodic measure $m$. However, there are simple examples of systems such as the Bernoulli shift which admit uncountably many distinct ergodic measures. Thus, for a given point, there may be uncountably many possibilities for its ergodic average. Indeed, its ergodic average may not exist at all. What can be said about where the Birkhoff sum 'goes'? How `large' is the set of points whose ergodic average takes a given value? I will explain some results in this direction in the case of symbolic dynamics and a variety of systems that admit codings. The results will require an explanation of the theory of equilibrium/ Gibbs measures in symbolic dynamics and a discussion of characteristics of 'dimension type'. I aim to make the talk accessible to as wide an audience as possible, so I will define precisely any technical terms that I use, including ergodicity, Birkhoff sum and the Bernoulli shift.